In the case of known thermal afterburning systems and methods of the initially mentioned type as are currently found on the market, the procedure is as follows: In the case of the maximum expected and permitted load of the exhaust air, the temperature is determined at which the clean air has the desired purity, in particular a specific maximum value of the carbon monoxide content. This temperature is then constantly regulated during the operation of the thermal afterburning system, regardless of the pollutant load that the supplied exhaust air actually has. Over long operating periods, this generally results in a higher temperature being produced in the combustion space of the combustion chamber than would really be necessary to achieve the desired purity in view of the currently prevailing pollutant load of the exhaust air.